Download Confidence Intervals with Means

Confidence
Intervals with
Means
Rate your confidence
0 - 100
Name my age within 10 years?

within 5 years?

within 1 year?

Shooting a basketball at a wading pool,
will make basket?
 Shooting the ball at a large trash can, will
make basket?
 Shooting the ball at a carnival, will make
basket?

What happens to your
confidence as the interval
gets smaller?
The larger your confidence,
the wider the interval.
Guess the number
Teacher will have pre-entered a number
into the memory of the calculator.
 Then, using the random number generator
from a normal distribution, a sample mean
will be generated.
 Can you determine the true number?

Point Estimate
 Use
a single statistic based on
sample data to estimate a
population parameter
 Simplest approach
 But not always very precise due to
variation in the sampling
distribution
Confidence intervals
 Are
used to estimate the
unknown population mean
 Formula:
estimate + margin of error
Margin of error
Shows how accurate we believe our
estimate is
 The smaller the margin of error, the
more precise our estimate of the true
parameter
 Formula:

Confidence level
 Is
the success rate of the method
used to construct an interval that
contains that true mean
 Using this method, ____% of the
time the intervals constructed
will contain the true population
parameter
What does it mean to be
95% confident?
 95%
chance that m is contained
in the confidence interval
 The probability that the interval
contains m is 95%
 The method used to construct
the interval will produce
intervals that contain m 95% of
the time.
Critical value (z*)
Found from the confidence level
 The upper z-score with probability p
lying to its right under the standard
normal curve

z*=1.645
z*=1.96
z*=2.576z*
Confidence level tail area
90%
95%
99%
.05
.025
.005
1.645
.05
.025 1.96
.005
2.576
Confidence interval for a
population mean:
Critical
value
Standard Error
(SD of
Margin of error Sampling
Distribution)
estimate
CI of Means - Steps:
Assumptions –
• SRS from population
• Sample is independent – smaller than
10% of population
• Sampling distribution is normal (or
approximately normal)
 Given (normal)
 Large sample size (approximately
normal)
 Graph data (approximately normal)
• σ is known
2) Calculate the interval
1)
3) Write a statement about the
interval in the context of the problem.
Statement: (memorize!!)
We are __________%
confident that the true
mean context lies
within the interval
_______ and ______.
Confidence Interval Applet

http://bcs.whfreeman.com/tps4e/#62864
4__666391__

The purpose of this applet is to
understand how the intervals move but
the population mean doesn’t.
A test for the level of potassium in the blood is
not perfectly precise. Suppose that repeated
measurements for the same person on different
days vary normally with σ = 0.2. A random
sample of three has a mean of 3.2. What is a
90% confidence interval for the mean potassium
level?
Assumptions:
Have an SRS of blood measurements
Assume patient has time to regenerate blood for
each sample so each sample is independent
Potassium level is normally distributed (given)
s known
We are 90% confident that the true mean
potassium level is between 3.01 and 3.39.
95% confidence interval?
Assumptions:
Assume same as previous
We are 95% confident that the true mean
potassium level is between 2.97 and 3.43.
99% confidence interval?
Assumptions:
Assume same as previous
We are 99% confident that the true mean
potassium level is between 2.90 and 3.50.
What happens to the interval as the
confidence level increases?
the interval gets wider as the
confidence level increases
How can you make the margin
of error smaller?

z* smaller
(lower confidence level)

σ smaller
(less variation in the population)

Really cannot
n larger
change!
(to cut the margin of
error in half,
n must be 4 times as big)
A random sample of 50 CHS
students was taken and their mean
SAT score was 1250. (Assume σ =
105) What is a 95% confidence
interval for the mean SAT scores of
CHS students?
We are 95% confident that the true
mean SAT score for CHS students is
between 1220.9 and 1279.1
Suppose that we have this
random sample of SAT scores:
950 1130 1260 1090 1310 1420 1190
What is a 95% confidence interval for
the true mean SAT score? (Assume s
= 105)
We are 95% confident that the true
mean SAT score for CHS students is
between 1115.1 and 1270.6.
Find a sample size:

If a certain margin of error is
wanted, then to find the sample size
necessary for that margin of error
use:
Always round up to the nearest person!
The heights of CHS male students
is normally distributed with σ =
2.5 inches. How large a sample is
necessary to be accurate within
+/- .75 inches with a 95%
confidence interval?
n = 43